Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

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June 2015, 5(2): 359-376. doi: 10.3934/mcrf.2015.5.359

Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

Guangliang Zhao ^1, , Fuke Wu ^2, and George Yin ^3,

1.	GE Global Research, 1 Research Circle, Niskayuna, NY 12309, United States
2.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074
3.	Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Received January 2014 Revised February 2014 Published April 2015

Abstract
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To treat networked systems involving uncertainty due to randomness with both continuous dynamics and discrete events, this paper focuses on diffusions modulated by a continuous-time Markov chain. In our paper [19], we considered ordinary differential equations with Markovian switching. This paper further treats more complex cases, namely, stochastic differential equations with Markovian switching. Our goal is to stabilize the systems under consideration. One of the difficulties is that the systems grow much faster than the allowable rates in the literature of stochastic differential equations. As a result, the underlying systems have finite explosion time. To overcome the difficulties, we develop feedback controls to extend the local solutions to global solutions and to stabilize the resulting systems. The feedback controls are Brownian type of perturbations. We establish the existence of global solution, prove the stability of the resulting systems, obtain boundedness in probability as $t\to\infty$, and provide sufficient conditions for almost sure stability. Then we present numerical examples to illustrate the main results.

Keywords: stabilization, feedback control., regularity, stability, Regime-switching diffusion.

Mathematics Subject Classification: Primary: 60J60, 60J27, 93E15; Secondary: 93E0.

Citation: Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control and Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359

References:

[1]	J. A. D. Appleby and X. Mao, Stochastic stabilisation of functional differential equations, Systems & Control Letters, 54 (2005), 1069-1081. doi: 10.1016/j.sysconle.2005.03.003.
[2]	J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control, 53 (2008), 683-691. doi: 10.1109/TAC.2008.919255.
[3]	L. Arnold, H. Crauel and V. Wihstusz, Stabilization of linear system by noise, SIAM J. Control Optim., 21 (1983), 451-461. doi: 10.1137/0321027.
[4]	A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380. doi: 10.1016/j.jmaa.2003.12.004.
[5]	F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth, Systems & Control Letters, 57 (2008), 262-270. doi: 10.1016/j.sysconle.2007.09.002.
[6]	M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), 1952-1988. doi: 10.1137/S0363012996299302.
[7]	R. Z. Khasminskii, Stochastic Stability of Differential Equations, $2^{nd}$ edition, Springer, Berlin, 2012. doi: 10.1007/978-3-642-23280-0.
[8]	R. Z. Khasminskii and G. Yin, Asymptotic behavior of parabolic equations arising from null-recurrent diffusions, J. Differential Eqs., 161 (2000), 154-173. doi: 10.1006/jdeq.1999.3647.
[9]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, $2^{nd}$ edition, Springer-Verlag, New York, NY, 2003. doi: 10.1007/b97441.
[10]	B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339 (2008), 419-428. doi: 10.1016/j.jmaa.2007.06.058.
[11]	R. Liptser and A. N. Shiryaev, Theory of Martingale, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2438-3.
[12]	X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2.
[13]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006. doi: 10.1142/9781860948848_fmatter.
[14]	X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, UK, 2008. doi: 10.1533/9780857099402.
[15]	A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Amer. Math. Soc., Providence, RI, 1989.
[16]	F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control, 82 (2009), 2150-2157. doi: 10.1080/00207170902968108.
[17]	G. Yin, X. R. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: Numerical algorithms and existence and uniqueness of solutions, SIAM J. Math. Anal., 41 (2010), 2335-2352. doi: 10.1137/080727191.
[18]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.
[19]	G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382. doi: 10.1137/110851171.
[20]	C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170. doi: 10.1016/j.jmaa.2009.03.066.

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References:

[1]	J. A. D. Appleby and X. Mao, Stochastic stabilisation of functional differential equations, Systems & Control Letters, 54 (2005), 1069-1081. doi: 10.1016/j.sysconle.2005.03.003.
[2]	J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control, 53 (2008), 683-691. doi: 10.1109/TAC.2008.919255.
[3]	L. Arnold, H. Crauel and V. Wihstusz, Stabilization of linear system by noise, SIAM J. Control Optim., 21 (1983), 451-461. doi: 10.1137/0321027.
[4]	A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380. doi: 10.1016/j.jmaa.2003.12.004.
[5]	F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth, Systems & Control Letters, 57 (2008), 262-270. doi: 10.1016/j.sysconle.2007.09.002.
[6]	M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), 1952-1988. doi: 10.1137/S0363012996299302.
[7]	R. Z. Khasminskii, Stochastic Stability of Differential Equations, $2^{nd}$ edition, Springer, Berlin, 2012. doi: 10.1007/978-3-642-23280-0.
[8]	R. Z. Khasminskii and G. Yin, Asymptotic behavior of parabolic equations arising from null-recurrent diffusions, J. Differential Eqs., 161 (2000), 154-173. doi: 10.1006/jdeq.1999.3647.
[9]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, $2^{nd}$ edition, Springer-Verlag, New York, NY, 2003. doi: 10.1007/b97441.
[10]	B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339 (2008), 419-428. doi: 10.1016/j.jmaa.2007.06.058.
[11]	R. Liptser and A. N. Shiryaev, Theory of Martingale, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2438-3.
[12]	X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2.
[13]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006. doi: 10.1142/9781860948848_fmatter.
[14]	X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, UK, 2008. doi: 10.1533/9780857099402.
[15]	A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Amer. Math. Soc., Providence, RI, 1989.
[16]	F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control, 82 (2009), 2150-2157. doi: 10.1080/00207170902968108.
[17]	G. Yin, X. R. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: Numerical algorithms and existence and uniqueness of solutions, SIAM J. Math. Anal., 41 (2010), 2335-2352. doi: 10.1137/080727191.
[18]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.
[19]	G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382. doi: 10.1137/110851171.
[20]	C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170. doi: 10.1016/j.jmaa.2009.03.066.

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